Question: Simplify; express your answer in exponential form. Assume $x\neq 0, r\neq 0$. $\dfrac{{(x^{-3}r^{5})^{-2}}}{{(x^{2}r^{-4})^{5}}}$
Explanation: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(x^{-3}r^{5})^{-2} = (x^{-3})^{-2}(r^{5})^{-2}}$ On the left, we have ${x^{-3}}$ to the exponent ${-2}$ . Now ${-3 \times -2 = 6}$ , so ${(x^{-3})^{-2} = x^{6}}$ Apply the ideas above to simplify the equation. $\dfrac{{(x^{-3}r^{5})^{-2}}}{{(x^{2}r^{-4})^{5}}} = \dfrac{{x^{6}r^{-10}}}{{x^{10}r^{-20}}}$ Break up the equation by variable and simplify. $\dfrac{{x^{6}r^{-10}}}{{x^{10}r^{-20}}} = \dfrac{{x^{6}}}{{x^{10}}} \cdot \dfrac{{r^{-10}}}{{r^{-20}}} = x^{{6} - {10}} \cdot r^{{-10} - {(-20)}} = x^{-4}r^{10}$